Risk management under the SABR model

Indications regarding the use of SABR in daily risk-management under either the Black or the Normal variant

Risk management of interest-rate derivatives can lead to a variety of definitions for each of the Greeks. In this article we examine the differences brought by each of the choices. We also discuss the difference between the Normal and Black calibration spaces.

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The SABR model owes its popularity to the fact that it can reproduce comparatively well the market-observed volatility smile and that it provides a closed-form formula for the implied volatility. In fact, because of these two features most practitioners use the SABR model mostly as a smile-interpolation tool rather than a pricing tool.

The SABR model is not the first smile-interpolation tool. Market dealers have been trying to fit the smile using a variety of models, ranging from plain statistical regressions across the quoted market points to other semi-rigorous approaches, such as the Vanna-Volga method in FX. Although most of these attempts manage to successfully fit the market smile, they do not provide a self-consistent arbitrage–free solution between pricing vanillas (using the smile) and pricing exotics (using Monte Carlo). On the contrary, the SABR model is setup on a coupled set of stochastic differential equations and can thus, in theory, answer both the need for smile-interpolation and for Monte-Carlo pricing. Theory versus practice, however, can often be miles apart and SABR is not an exception.

The reason is that one of SABR’s main strengths is also one of its key weaknesses. The treasured closed-form formula for the implied vol that offers the possibility for quick calibrations and quick quotes is based on an approximation which breaks down for large strikes and large maturities. The fact that this approximation breaks down implies that vanilla versus non-vanilla pricing (analytic formula pricing versus Monte-Carlo pricing) would not be consistent anymore.

The approximation works well for reasonable choices of strikes and maturities offering the possibility to manage and hedge market risks, such as the Delta, Vega, Gamma, etc. In some practices, the daily risk-management goes even further by managing the sensitivities associated to the four SABR parameters.

The SABR formula can be found in two variants: the Black SABR formula or the Normal SABR formula, which, respectively, express Black or Bachelier implied vols in terms of the SABR parameters (this is commonly termed the Black or Normal calibration space).

In recent times, SABR has been brought back to attention due to the negative-rate environment. In this environment a solution is required for quoting volatilities at a negative strike. Currently, two mainstream approaches have emerged: (i) the classical SABR model in the Black calibration space is modified by introducing a shift and (ii) the classical SABR model is maintained but it is only used in the special case of beta=0 and under the Normal calibration space.

In this article we provide some indications regarding the use of SABR in daily risk-management under either the Black or the Normal variant.

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